## $$C$$-totally real submanifolds of $$\mathbb{R}^{2n+1}$$ satisfying a certain inequality.(English)Zbl 1029.53064

In theorem 1 the author establishes a sharp inequality between the squared mean curvature $$||H||^2$$ and the scalar curvature $$\tau$$ for a $$C-$$totally real submanifold of maximum dimension in a Sasakian space form. He also investigates the equality case in the following:
Theorem 2. Let $$i:M^n\rightarrow \mathbb{R}^{2n+1}$$ be a $$C$$-totally real immersion satisfying the equality $$||H||^2={{2(n+2)}\over{n^2(n-1)}}\tau$$. Then either $$M$$ is a totally geodesic submanifold and hence locally isometric to the real space $$\mathbb{R}^n$$ or the set $$U$$ of non-totally geodesic points in $$m$$ is a dense subset of $$M$$, $$U$$ is an open portion of a Whitney sphere $$\widetilde w(S^n)$$ with $$a>1$$ and, up to rigid body motions of $$\mathbb{R}^{2n+1}$$, the immersion $$i$$ is given by $$\widetilde w$$, where $$\widetilde w:S^n\rightarrow \mathbb{R}^{2n+1}$$ is the immersion lifted from the Whitney immersion. The map $$f:E^{n+1}\rightarrow \mathbb{C}^{n}$$ from Euclidean space into the complex Euclidean space $$\;f(x_0,x_1,\dots,x_n)={{1}\over{1+x_0^2}}(x_1,\dots,x_n,x_0x_1,\dots,x_0x_n)$$ restricted on the unit hypersphere $$S^n$$ is called the Whitney immersion.

### MSC:

 53C40 Global submanifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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